Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

Topological Photonics

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on their surfaces. In two dimensions, electrons on the surface of a topological insulator are not scattered despite defects and disorder, providing robustness akin to that of superconductors. Topological insulators are predicted to have wide-ranging applications in fault-tolerant quantum computing and spintronics. Substantial effort has been directed towards realizing topological insulators for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. But because magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatter-free edge states requires a fundamentally different mechanism-one that is free of magnetic fields. A number of proposals for photonic topological transport have been put forward recently. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, in which temporal variations in solid-state systems induce topological edge states. Here we propose and experimentally demonstrate a photonic topological insulator free of external fields and with scatter-free edge transport-a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrodinger equation where the propagation coordinate (z) acts as 'time'. Thus the helicity of the waveguides breaks z-reversal symmetry as proposed for Floquet topological insulators. This structure results in one-way edge states that are topologically protected from scattering.

Photonic Floquet topological insulators

Artificial Neural Networks have dramatically improved performance for many machine learning tasks. We demonstrate a new architecture for a fully optical neural network that enables a computational speed enhancement of at least two orders of magnitude and three orders of magnitude in power efficiency over state-of-the-art electronics.

Deep learning with coherent nanophotonic circuits

In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. This Review Article outlines the exploration of the interplay between parity–time symmetry and non-Hermitian physics in optics, plasmonics and optomechanics.

Non-Hermitian physics and PT symmetry

We present the first experimental observation of a topological transition in a non-Hermitian system. In contrast to standard methods for examining topological properties, which involve probing edge (or surface) states, we monitor the topological transition by employing bulk dynamics only. The system is composed of a lattice of evanescently coupled optical waveguides, and non-Hermitian behavior is engineered by inducing bending loss by spatially "wiggling" every second waveguide.

Observation of a Topological Transition in the Bulk of a Non-Hermitian System.

Topological insulators are a new class of materials that exhibit robust and scatter-free transport along their edges - independently of the fine details of the system and of the edge - due to topological protection. To classify the topological character of two-dimensional systems without additional symmetries, one commonly uses Chern numbers, as their sum computed from all bands below a specific bandgap is equal to the net number of chiral edge modes traversing this gap. However, this is strictly valid only in settings with static Hamiltonians. The Chern numbers do not give a full characterization of the topological properties of periodically driven systems. In our work, we implement a system where chiral edge modes exist although the Chern numbers of all bands are zero. We employ periodically driven photonic waveguide lattices and demonstrate topologically protected scatter-free edge transport in such anomalous Floquet topological insulators.

/pdf/observation-of-photonic-anomalous-floquet-topological-11ljmple8c.pdf

Observation of photonic anomalous Floquet topological insulators

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs
only on the surface. In two dimensions, surface electrons in topological insulators do not scatter despite defects
and disorder, providing robustness akin to superconductors. Topological insulators are predicted to have wideranging
applications in fault-tolerant quantum computing and spintronics. Recently, large theoretical efforts were
directed towards achieving topological insulation for electromagnetic waves. One-dimensional systems with
topological edge states have been demonstrated, but these states are zero-dimensional, and therefore exhibit no
transport properties. Topological protection of microwaves has been observed using a mechanism similar to
the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. However,
since magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatterfree
edge states requires a fundamentally different mechanism - one that is free of magnetic fields. Recently, a
number of proposals for photonic topological transport have been put forward. Specifically, one suggested
temporally modulating a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge
states. This is in the spirit of the proposed Floquet topological insulators, where temporal variations in solidstate
systems induce topological edge states. Here, we propose and experimentally demonstrate the first external
field-free photonic topological insulator with scatter-free edge transport: a photonic lattice exhibiting topologically
protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled
helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by
a Schrodinger equation where the propagation coordinate acts as ‘time’. Thus the waveguides' helicity breaks zreversal
symmetry in the sense akin to Floquet Topological Insulators. This structure results in scatter-free, oneway
edge states that are topologically protected from scattering.

Magnetic effects are fundamentally weak at optical frequencies. Now, by applying inhomogeneous strain in photonic band structures of a honeycomb lattice of waveguides, scientists show experimentally and theoretically that it is possible to induce a pseudomagnetic field at optical frequencies. The field yields 'photonic Landau levels', which suggests the possibility of achieving greater field enhancements and slow-light effects in aperiodic photonic crystal structures than those available in periodic structures.

Julia M. Zeuner

Papers

Photonic Floquet topological insulators

Observation of a Topological Transition in the Bulk of a Non-Hermitian System.

Observation of photonic anomalous Floquet topological insulators

Photonic Floquet topological insulators

Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures